274 
MR. A. CAYLEY’S ANALYTICAL RESEARCHES CONNECTED WTTH 
■2v/afi(v/aiB-®)(a«*+..|5‘)={(\/g£-®)(Va-(3yB)+'/(ex/s-s>/c)}’, 
2s/a$(v'ac-©)(ac«'“+..|5*)={(N/a®-<©)(«\/9-yv/€)+/3(eyc-jfv/g)}''. 
whence 
-^■y2v<gS(s/ai-®)(g«+s/3+igy)= 
(^/9l-®)^/a»-(^/as-®)^/S|3+((g^/5-Jr^/a)'/ 
^\/-2s/M{\/X-iB) (aa+®/3+ ®y) = 
(x/9C-©)N/a«+(i!x/C-4Ps/a)P-(-ya£-®)r 
cj3’ + bf — 2//3y + = 0, 
and putting for a moment 
l^=\/ — 2\/ 9iC(\/ ^C — (§) 
v = \/ n/ 2\/^B(\/ — ^). 
After some reductions, and observing that the ratios oniy of the quantities a, j3, y. t 
are material, 
and it is easily seen also that the coordinates of the point of contact are 
also 
And substituting and introducing throughout the quantities f, g, h, J, also forming 
the analogous equations, the equations of the tactors are 
P(-P+g>+!.>) + (g+h)j(f’-(g-hf-?®)|-;;;L,t 
- f 
— I p-(g-hf-?s!i(£-!i 
1j 
J s/^\ 
", g\ 
1 li / 
" h\ 
r 
.'-j) 
